New approach to twisted q-Bernoulli polynomials

respectively. In addition, the Bernoulli numbers are given by Bn := Bn() for n ≥ . Recently, the Bernoulli polynomials and Bernoulli numbers have gained considerable significance in the fields of physics and mathematics [–]. For example, Kim [] defined a new q-analogy of the Bernoulli polynomials and Bernoulli numbers, and he deduced some important relations between them. Moreover, q-analogues have been investigated in the study of quantum groups and q-deformed superalgebras []. The connection here is similar, in that much the string theory is set in the language of Riemann surfaces, resulting in connections with elliptic curves, which in turn relate to q-series. A q-analogue is an identity for a q-series that returns a known result in the ‘bosonic’ limit (in contrast to the conventional ‘fermionic’ limit q → –) as q →  (from inside the complex unit circle in most situations). In addition to the widely used q-series, we have q-numbers, q-factorials, and q-binomial coefficients. A q-number is obtained by observing limq→ –q n –q = n. Thus, we define a q-number as [n]q = –q n –q . Accordingly, one can define the q-analogue of the factorial, namely, q-factorial, as


Introduction
The classical Bernoulli polynomials B n (x) and the Euler polynomials E n (x) are usually defined by the generating functions te xt e t - = ∞ n= B n (x) t n n! , |t| < π and e xt e t +  = ∞ n= E n (x) t n n! , |t| < π, respectively. In addition, the Bernoulli numbers are given by B n := B n () for n ≥ . Recently, the Bernoulli polynomials and Bernoulli numbers have gained considerable significance in the fields of physics and mathematics [-]. For example, Kim [] defined a new q-analogy of the Bernoulli polynomials and Bernoulli numbers, and he deduced some important relations between them. Moreover, q-analogues have been investigated in the study of quantum groups and q-deformed superalgebras []. The connection here is similar, in that much the string theory is set in the language of Riemann surfaces, resulting in connections with elliptic curves, which in turn relate to q-series. A q-analogue is an identity for a q-series that returns a known result in the 'bosonic' limit (in contrast to the conventional 'fermionic' limit q → -) as q →  (from inside the complex unit circle in most situations). In addition to the widely used q-series, we have q-numbers, q-factorials, and q-binomial coefficients. A q-number is obtained by observing lim q→ -q n -q = n. Thus, we define a q-number as [n] q = -q n -q . Accordingly, one can define the q-analogue of the factorial, namely, q-factorial, as [n] q ! = [] q [] q · · · [n -] q [n] q = q q q  q · · · q n q =  · ( + q) · · ·  + q + · · · + q n-  + q + · · · + q n- .  licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.advancesindifferenceequations.com/content/2013/1/298 Using this notation, we can define the q-binomial coefficients, also known as Gaussian coefficients, by Furthermore, the q-Bernoulli polynomials β n,q (x) and the q-Bernoulli numbers β n,q can be defined in terms of the generating function F t (x, q) as follows []: Kim [] established an interesting relation between Bernoulli numbers and q-integers, that is, In addition, Kim [, Theorem ], Kim and Lee [, Lemma .] derived the relations between the Euler polynomials E (r) n (x) of order r using the alternating sum of powers of consecutive integers T k (n). Here, T k (n) = n i= (-) r l k and E (r) n (x) is defined as Simsek constructed twisted Bernoulli polynomials together with twisted Bernoulli numbers and obtained analytic properties of twisted L-functions [, ]. Further, he defined generating functions of the twisted q-Bernoulli numbers and polynomials []. In a complex case, the generating function of twisted q-Bernoulli numbers f q,ω (t) and a q-analogue of the Hurwitz zeta function f ω (t, x, q) are given by where q ∈ C with |q| < , and ω is the rth root of . In a complex case, the generating function of twisted q-Bernoulli numbers f q,ω (t) and a q-analogue of the Hurwitz zeta function f ω (t, x, q) are given by t l l! , http://www.advancesindifferenceequations.com/content/2013/1/298 where q ∈ C with |q| < , and ω is the rth root of . Simsek [] then derived the identities where , and x is a natural number. In this paper, we first study relations among q-consecutive integers, q-Bernoulli numbers, and q-Euler numbers.
In , Faulhaber [] evaluated the sums of powers of consecutive integers  k + · · · + n k up to k = . Further, in , Knuth [] presented an insightful alternative account of Faulhaber's work. Several mathematicians further considered the problems of q-analogues of such sums of powers [-, ]. On the basis of Bernoulli's concept, Kim derived a qanalogue of the sums of powers of consecutive integers, by setting with s ∈ C and |s| < .
In Section , we recall some necessary identities for basic hypergeometric series []. Further, we obtain a generalization of Proposition ., and accordingly, we obtain qconsecutive integers for n- k= [k] q q k . These new results are similar to the ones presented in some other studies [-] and [].
In Section , we derive a formula for S ,n and T ,q (n) by using a property of basic hypergeometric series, such as The q-analogue Eulerian numbers are defined as []: For these, we establish certain new identities by utilizing basic hypergeometric series, which differ from Bernoulli numbers and polynomials constructed by Kim et al. [, ] as follows: if n is even, Here, we note that these are related to T l,q h (n).
In Section , we deduce recursive formulas from Lemma . for basic hypergeometric series. More precisely, let S l (t) = ∞ k= (q  ;q) l k (q;q) l k t k . Then, we derive the recursive formu- Using these identities, we obtain a formula for n- k= [k] l q k , and we present relations between q-Bernoulli numbers and q-consecutive integers, which are related to (S)-(S). Lastly, the rank of partition is defined as the difference between its largest part and the number of its parts. The number of partitions of n with the rank r would be denoted by P r (n). We use the convention P  () = , P r (n) =  for r = , n ≤  and r = , n < . Here, for the sake of convenience, we define Then, these are related to P r (n) by the following identity (Remark .): with |q|  < |u| < . Finally, we shall relate through Theorem . and Remark ., q-Bernoulli polynomials with the third-order mock theta functions introduced by Ramanujan. Throughout this paper, we adopt the following notations: • ω: the rth root of unity.

Identities of basic hypergeometric series and n-1 k=0 [k] q q k
In this section, we investigate some identities of basic hypergeometric series. To this end, we refer to []. Now, we consider the series defined by Fine presented many interesting properties in his book; the following identity represents one such property: Throughout this paper, q denotes a fixed complex number of absolute value less than , so that we may write q = exp(πiτ ), where τ is a complex number with a positive imaginary part. We use q c to denote exp(cπiτ ). The partial product (aq; q) n converges for all values of a, as may be easily seen from the absolute convergence of q n . Hence, if b is not one of the values q - , q - , . . . , the coefficients (aq;q) n (bq;q) n are bounded, and the series (.) converges for all t inside the unit circle, and represents an analytic function therein. Hence, the function on the right-hand of (.) is regular in the domain |t| < |q| - , except for a simple pole at t = . Therefore, we obtain the continuation of F to a larger circle. Then, it is easy to apply (.) again to the continuation of F to the circle |t| < |q| - , and thus, we conclude that for b = q -n , n > , the only possible singularities of F occur at the points t = q -n (n ≥ ), which are simple poles in general. As a function of b, F is regular, except possibly at the simple poles b = q -n (g ≥ ), provided that b and t do not have one of the singular values mentioned above. First, we derive Theorem . by generalizing the following proposition.
Proposition . For the complex number q and t with |q| < , we have To prove this, we need some identities from [].

Lemma . () For a nonnegative integer N,
It is an analogue of the binomial series, to which one can reduce termwise with q = .
Proof of Theorem . We start with the left-hand side in our assertion: Replacing t by aq lm in Lemma .(), we claim that (bq k+ ; q) n (q; q) n t n q kln . http://www.advancesindifferenceequations.com/content/2013/1/298 By substituting bq k and tq kl for a and t, respectively, in Lemma .(), we derive Since (a; q) n = (a;q) ∞ (aq n ;q) ∞ , it follows that the last can be written as Thus, we deduce the identity as desired.
Next, we present alternative proofs of the following results of Kim [] as an application of Theorem ..
Note that it is exactly the same as (.).
These are the q-analogues of n k= k = n+  .
Proof () Replacing both b and l by  in Theorem ., we see that After substituting at for a in Lemma .(), if we apply it to the above, we get Putting a = q in the above, we get . http://www.advancesindifferenceequations.com/content/2013/1/298 However, by using the notation defined in Section , the left-hand side of the above can be written as Thus, from the calculations above and the fact that By considering the exponent of t, we conclude that () If we put t = q in (), we have the first equality On the other hand, a direct calculation gives Therefore, we establish the claim. () It follows from () that Moreover, if we replace n - by n in () and multiply both sides by q n , we obtain Observe that the identity above with t = q - turns out to be a Warnaar's identity []: Finally, on the basis of geometric series, we get When t = q - in the above, we have Note that this formula was also derived by several mathematicians such as Schlosser [ Thus, by combining (.), (.), and (.), we reach the conclusion.

q-Consecutive and q-analogue of Eulerian numbers E 1,q and E 2,q
We have studied the infinite sum with linear coefficients for q-numbers in the previous section. In this section, we consider the sum with quadratic coefficients, i.e., the following equation.

Theorem . As a finite sum for q, we get
Proof Putting t = q in Lemma ., we derive by the definition of [n] q . Therefore, the last equality follows. .
The other cases l >  will be studied in greater detail in the next section. This was previously proved by Schlosser [] by using Bailey's terminating very-well-poised balanced  φ  transformation.
Proof By definition in Section , we see that Then, the sum of formulas after setting t = q -  and t = q -  shows that our corollary is true. Carlitz [] constructed a q-analogue of Eulerian numbers. On the other hand, Kim considered the following functions []: and For m, n ∈ N, he showed that [, Proposition ] A similar result is in [, Lemma .]. Thus, we get the results for l =  and  as follows.
if n is even, ) if n is even, Proof Replacing t by -t in Corollary .(), we see that If we let t = q, it becomes if n is even, Comparing this with (.), we can prove (). As for (), if we substitute -q with t in Lemma ., it turns out that  -(-q  ) n  + q  . http://www.advancesindifferenceequations.com/content/2013/1/298 For an even integer n, the above becomes Similarly, for an odd integer n, we get the result Remark . In the proof above, as in the case of (.), we can obtain an equation by plugging -q into t: Further, we have  n- k= [k] q q k by adding (.) and (.). Indeed, Note that this can be written as [∞] q ( [] q ) in terms of q-number notation. Alternatively, it may be factorized and expressed as from which we derive Remark . In [, Theorem ], Kim derived a summation formula for E m,q , We also see one for E k,q (x),

Difference equation and q-consecutive integer
As mentioned in Theorem ., in this section, we study ∞ k= (q  ;q) l k (q;q) l k t k for more general cases l and its similar sum with q-binomial coefficients. In addition, we show the relations between these and twisted q-Bernoulli numbers. To this end, we need the following lemma.

Lemma . Given a sequence
Proof See Section , [].
Using this lemma, we generalize the identities considered in the previous two sections.

Proposition . For a positive integer l, we have the identities
If l is  (respectively, ), this would be the result of Corollary .() (respectively, Lemma .).
Proof Let S l (t) be a series defined by ∞ k= (q  ;q) l k (q;q) l k t k for a nonnegative integer l. By setting a = q, b = , A k = (q  ;q) l k (q;q) l k , and g(t) = S l (t) in Lemma ., we derive the following recursive formula: Multiplying both sides by t, we get Further, by induction, Considering tS  (t) = t ∞ k= t k = t -t , we are able to rewrite the above as , http://www.advancesindifferenceequations.com/content/2013/1/298 and All the denominators on the right-hand side are factorized as l +  terms. However, the numerators are somewhat complex. Therefore, we recursively define a sequence C l (t; q) with l ≥  as follows: Then, we get the following theorem.

Theorem . ()
The infinite sum T l,t is expressed as a quotient of C l (t; q) by l +  products, precisely speaking, Replacing t by  t in Theorem .(), we can deduce one of Simsek's relations [, Proposition .].

Theorem . The generating function (complex cases) of twisted q-Bernoulli numbers is given by
where ω is the rth root of unity and Proof If we recall (S) from Section , we get B * ,ω (q) = , Since we know from [, (.)] that F(a, ; t : q) = (atq : q) ∞ (t : q) ∞ , by setting q and t to be a and ω - q - , respectively, we obtain When l is greater than , we get By (S) and Theorem ., we get a corollary. Substituting l -, x + , and ω - q -l for l, n, and t in (.), respectively, we establish the last identities.

Corollary . If l is an integer greater than , we have
As its immediate corollary, we have the following. Moreover, we can deduce the following corollary, which is analogous to Theorem ..

Proposition . For a nonnegative integer n,
∞ k= k + n n q t k =  (t; q) n+ , and when n = , we get the following by considering the summation from  to n - in the above: () n- k= k +   q q k = n +   q . http://www.advancesindifferenceequations.com/content/2013/1/298 By letting u = - on the right-hand side, we get Further, it gives rise to a third-order mock theta function f (q) =  + ∞ n= α(n)q n =  +